Variable-precision distributed arithmetic multi-input multi-output equalizer for power-and-area-efficient optical dual-polarization quadrature phase-shift-keying system

ABSTRACT

A variable-precision distributed arithmetic (VPDA) multi-input multi-output (MIMO) equalizer is presented to reduce the size and dynamic power of 112 Gbps dual-polarization quadrature phase-shift-keying (DP-QPSK) coherent optical communication receivers. The VPDA MIMO equalizer compensates for channel dispersion as well as various non-idealities of a time-interleaved successive approximation register (SAR) based analog-to-digital converter (ADC) simultaneously by using a least mean square (LMS) algorithm. As a result, area-hungry analog domain calibration circuits are not required. In addition, the VPDA MIMO equalizer achieves 45% dynamic power reduction over conventional finite impulse response (FIR) to equalizers by utilizing the minimum required resolution for the equalization of each dispersed symbol.

BACKGROUND OF THE INVENTION

1. Field of the Invention

Exemplary embodiments of the present invention relate to a variable-precision distributed arithmetic (VPDA) multi-input multi-output (MIMO) equalizer for power- and area-efficient 112 Gb/s optical dual-polarization quadrature phase-shift-keying (DP-QPSK) system.

2. Discussion of the Background

Coherent optical dual-polarization quadrature phase-shift-keying (DP-QPSK) systems with electrical domain dispersion compensation at a wavelength of 1550 nm are being adapted for 112 Gb/s long-haul optical communication links in order to combat chromatic dispersion (CD). FIG. 1 shows a typical block diagram of a 11-Gb/s DP-QPSK coherent optical receiver. The phase information of the received optical signal is converted to two pairs of analog voltage signals and 2× oversampled using four 56 GS/s analog-to-digital converters (ADCs). A digital equalizer subsequent to the ADCs compensates for the channel dispersion. Such a high data rate requires a massive parallelization in the ADC and digital equalizer due to the bandwidth limitation of active components. The tight performance requirements of the ADC necessitates complex and area-hungry calibration circuits to overcome various non idealities caused by mismatch among parallel ADCs and nonlinearities in each ADC. The power consumption of the digital equalizer increases in proportion to its dispersion compensation capability and the level of computational precision. The power and area of the conventional coherent DP-QPSK system have been the major impediments to its adaptation to high-volume applications such as Metro dense wavelength division multiplexing (DWDM) despite its excellent dispersion compensation capability.

SUMMARY OF THE INVENTION

An exemplary embodiment of the present invention discloses a variable-precision distributed arithmetic (VPDA) multi-input multi-output (MIMO) equalizer connected to outputs of a plurality of analog-to-digital converters (ADCs) based on time-interleaved successive approximation registers, the VPDA MIMO equalizer comprises a plurality of sub-equalizers classified into a first sub-equalizers group and a second sub-equalizers group, wherein each of sub-equalizers included in the first sub-equalizers group is connected to outputs of a first ADC group and each of sub-equalizers included in the second sub-equalizers group is connected to outputs of a second ADC group, and a decision unit configured to determine output signals using outputs of the plurality of sub-equalizers, wherein one sub-equalizer included in the first sub-equalizers group relates to other one sub-equalizer included in the second sub-equalizers group, and one output signal among the output signals is determined by the decision unit based on outputs of the one sub-equalizer and the other one sub-equalizer.

Each of the plurality of sub-equalizers comprises a plurality of additional-equalizers for distributed arithmetic, and each additional-equalizers corresponds to one of bits according to a resolution of the ADCs.

One additional-equalizer of the one sub-equalizer relates to other one additional-equalizer of the other one sub-equalizer, and the one output signal is determined by the decision unit based on an output of the one additional-equalizer and an output of the other one additional-equalizer.

Outputs of the ADCs corresponding to i-th bit of the bits is inputted to i-th additional-equalizers of the plurality of sub-equalizers.

Each of the plurality of additional-equalizers comprises a plurality of finite impulse responses (FIR) filters.

One part of the plurality of FIR filters is connected to outputs of one ADC of the first ADC group (or the second ADC group), and another part of the plurality of FIR filters is connected to outputs of other one ADC of the first ADC group (or the second ADC group).

The plurality of additional-equalizers configured to computing outputs according to a to sequence from the most significant bit (MSB) of the bits towards the least significant bit (LSB) of the bits.

The VPDA MIMO equalizer further comprises a range checker configured to determine whether further computing outputs for next bit of the sequence is required based on a equalized symbol determined by outputs for present bit and a decision threshold.

Coefficients with each of bits according to a resolution of the ADCs are combined with filter coefficients of FIR filters by at least one of a distributed arithmetic (DA) scheme and a least mean square (LMS) algorithm.

Each of outputs of the ADCs is multiplied with the combined coefficients and each of outputs of the FIR filters is combined.

An exemplary embodiment of the present invention discloses a dual-polarization quadrature phase-shift-keying (DP-QPSK) receiver comprises a plurality of analog-to-digital converters (ADCs) based on time-interleaved successive approximation registers, a variable-precision distributed arithmetic (VPDA) multi-input multi-output (MIMO) equalizer connected to outputs of the ADCs, wherein the VPDA MIMO equalizer comprises a plurality of sub-equalizers, and each of sub-equalizers comprises a plurality of additional-equalizers and a decision unit configured to determine output signals using outputs of the additional-equalizers.

It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are included to provide a further understanding of to the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention, and together with the description serve to explain the principles of the invention.

FIG. 1 shows a typical block diagram of a 112 Gbps DP-QPSK coherent optical receiver.

FIG. 2 shows the block diagram of the proposed coherent DP-QPSK receiver for 80 km metro DWDM applications according to an exemplary embodiment of the present invention.

FIG. 3 shows a conceptual block diagram of a 56 GS/s time-interleaved SAR-based ADC according to an exemplary embodiment of the present invention.

FIG. 4 shows a block diagram of the MEMO equalizer according to an exemplary embodiment of the present invention.

FIG. 5 shows a section of hardware utilized for the equalization of two transmitted symbols through X and Y polarizations according to an exemplary embodiment of the present invention.

FIG. 6 shows the block diagrams of one sub-equalizer in a conventional parallel FIR equalizer for 112 Gb/s DP-QPSK coherent systems according to an exemplary embodiment of the present invention.

FIG. 7 shows a block diagram of a five-bit switched-capacitor SAR-ADC according to an exemplary embodiment of the present invention.

FIG. 8 shows conceptual block diagram based on merging coefficients for nonlinearity compensation with those for MIMO equalization according to an exemplary embodiment of the present invention.

FIG. 9 shows a block diagram of a sub-equalizer of the DA MIMO equalizer according to an exemplary embodiment of the present invention.

FIG. 10 shows the equalized QPSK symbols under different ADC resolutions in a two-dimensional signal space according to an exemplary embodiment of the present invention.

FIG. 11 shows a conceptual block diagram of the proposed variable-precision architecture embedded in the DA framework according to an exemplary embodiment of the present invention.

FIG. 12 shows erroneous regions in the QPSK signal space when (1, 1) is sent according to an exemplary embodiment of the present invention.

FIG. 13 shows a simulation setup of the VPDA MIMO equalizer according to an exemplary embodiment of the present invention.

FIG. 14 shows randomly generated total 224 nonlinear characteristics of the SAR ADCs according to an exemplary embodiment of the present invention.

FIG. 15 shows BER increment with respect to the size of the suspicious regions for each stage according to an exemplary embodiment of the present invention.

FIG. 16 shows the simulated OSNR vs. BER graphs for three different equalizers according to an exemplary embodiment of the present invention.

FIG. 17 shows the simulated tracking performance of the VPDA MIMO equalizer under the rotation of the polarization plane at a rate of 50 rad/s according to an exemplary embodiment of the present invention.

DETAILED OF THE ILLUSTRATED ED EMBODIMENTS

The invention is described more fully hereinafter with reference to the accompanying drawings, in which exemplary embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these exemplary embodiments are provided so that this disclosure is thorough, and will fully convey the scope of the invention to those skilled in the art. In the drawings, the size and relative sizes of layers and regions may be exaggerated for clarity. Like reference numerals in the drawings denote like elements.

I. Introduction

In this paper, we propose a power-and-area-efficient variable-precision distributed arithmetic (VPDA) multi-input multi-output (MIMO) equalizer for coherent optical dual-polarization quadrature phase-shift-keying (DP-QPSK) systems suitable for Metro dense wavelength division multiplexing (DWDM) applications which require performance, low power and miniaturization. For example, the target distance may be 80 km. Significant reduction in power and area is achieved on the basis of the following two factors: (i) digital equalizer in this proposed design compensates for the channel dispersion as well as the non-idealities of the analog-to-digital converter (ADC) without hardware overhead, which does not necessitate area-hungry analog domain calibration circuits, and (ii) each dispersed symbol is equalized with the minimum required resolution. The latter factor leads to dynamic power reduction of 45% in the digital equalizer. Section II depicts the proposed receiver architecture and Section III describes the distributed arithmetic (DA) MIMO channel equalizer architecture which also compensates for various mismatches and non-linearities of the successive approximation register (SAR) ADC. Then, section IV describes the variable-precision concept applied to the DA MIMO architecture for the dynamic power reduction. Section V provides the simulation results of the VPDA MIMO equalizer and finally, section VI summarizes the discussion.

II. Receiver Architecture

FIG. 2 shows the block diagram of the proposed coherent DP-QPSK receiver for 80 km metro DAVDM applications according to an exemplary embodiment of the present invention. The outputs of four 56 GS/s 5 bit ADCs are connected to both the VPDA MIMO equalizer and a clock recovery block. Conventional dispersion-tolerant phase detectors typically operate under residual CD in long-haul coherent systems can be utilized in this case without pre-equalization because the target distance is only 80 km. In addition, because the phase detector detects the symbol rate directly from 2× oversampled ADC outputs without any assistant from the equalizer, typical start-up issue does not exist.

The VPDA MIMO equalizer is intentionally designed to operate in time domain because the frequency domain counterparts have higher level of complexity in 80 km metro applications where the chromatic dispersion is less than 1280 ps/nm.

III. Compensation of ADC Non-Ideality

Digital signal processing methods to overcome either the mismatch or nonlinearity have been reported. The proposed DA MIMO architecture integrates these two methods together with the channel equalizer without hardware overhead over conventional finite impulse response (FIR) channel equalizers employed in 112 Gb/s coherent optical DP-QPSK systems.

A. Mismatch Compensation

It has been shown that the MIMO equalization method can compensate for the offset, gain and sampling time mismatches of a time-interleaved parallel ADC. FIG. 3 shows a conceptual block diagram of a 56 GS/s time-interleaved SAR-based ADC according to an exemplary embodiment of the present invention. The resolution of the ADC is set to five bits to suppress the optical signal-to-noise ratio (OSNR) penalty below 0.5 dB at a bit error ratio (BER) of 10⁻³. In our ADC model, 16 track-and-hold (T/H) circuits operating at 3.25 GS/s are parallelized for the sampling rate of 56 GS/s. We assumed that the sub ADCs subsequent to the T/H requires seven clock cycles for each data conversion. Thus, a total of 16×7=112 SAR ADCs are parallelized. As a result, the digitized samples show 112 different cyclic characteristics, which can be modeled by a radix-112 system. Let denote the i-th input signal in the radix-112 system, then the n-th input matrix of the radix-112 ADC system may be given by the following Equation 1.

V _(I) [n]=[V _(I) ¹ [n]V _(I) ² [n] . . . V _(I) ¹¹² [n]] ^(T)  [Equation 1]

Similarly, the n-th output matrix of the ADC may be the following Equation 2.

V _(O) [n]=[V _(O) ¹ [n]V _(O) ² [n] . . . V _(O) ¹¹² [n]] ^(T)  [Equation 2]

Various mismatch effects of the ADC can be interpreted using linear MIMO matrices. Let the matrix G, P and O represent the gain, phase and offset mismatches of the time-interleaved ADC, respectively. V_(I) and V_(O) satisfy the following Equation 3.

[G] _(112×112) ·[P] _(112×112)Ø_(c) [V _(I)]_(112×1) +[O] _(112×1) =[V _(O)]_(112×1)  [Equation 3]

The mathematic operator

denotes the element-by-element convolution. The quantization noise is not considered for simplicity. The gain mismatch G is a diagonal matrix, which is given by the following Equation 4.

$\begin{matrix} {G_{ij} = \left\{ {\begin{matrix} g_{i} & {i = j} \\ 0 & {i \neq j} \end{matrix},i,{j = {1\mspace{14mu} \ldots \mspace{14mu} 112}}} \right.} & \left. {{Equation}\mspace{14mu} 4} \right\rbrack \end{matrix}$

wherein g_(i) represents the gain of the i-th ADC. The sampling phase mismatch P can be described using linear interpolation between adjacent samples in V_(I), as the following Equation 5.

P _(112×112)=Φ_(112×112) ⁻¹+Φ_(112×112) ⁰+Φ_(112×112) ¹Φ_(i,j) ^(k) [n]=φ _(i,j,k)δ[112n+i−j−k ], i,j=1 . . . 112  [Equation 5]

wherein φ_(i,j,k) is the phase coefficient for the i-th ADC input and δ[n] denotes the discrete time delta function. Because the input signal V_(I) is 2× oversampled, the P matrix models the phase mismatch with sufficient accuracy.

The offset vector O is a 112×1 matrix, as given by the following Equation 6.

O=[O ¹ O ² . . . O ¹¹²]^(T)  [Equation 6]

Wherein O^(i) represents the offset of the i-th ADC.

A coherent DP-QPSK receiver requires four 56 GS/s ADCs, as shown in FIG. 1; thus, the total number of sub ADCs is 112×4=448. Let S_(I) and S_(Q) be represented by the following Equation 7.

$\begin{matrix} {S_{I} = {{\begin{bmatrix} V_{I\; 1} \\ V_{I\; 3} \end{bmatrix} \cdot S_{Q}} = \begin{bmatrix} V_{I\; 2} \\ V_{I\; 4} \end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack \end{matrix}$

wherein S_(I) and S_(Q) denote the in-phase and quadrature-phase signals in a QPSK system, respectively. Equation 3 can be expanded to include all four ADC inputs S_(I) and S_(Q) as the following Equation 8.

M _(I)

S _(I) +j·M _(Q)

S _(Q) +C=R _(I) +j·R _(Q)  [Equation 8]

wherein M_(I) and M_(Q) may be represented by the following Equation 9. Also, C and R_(I)+j·R_(Q) may be represented by the following Equation 10.

$\begin{matrix} {{M_{I} = \begin{bmatrix} {G_{1} \cdot P_{1}} & 0 \\ 0 & {G_{3} \cdot P_{3}} \end{bmatrix}_{224 \times 224}}{M_{Q} = \begin{bmatrix} {G_{2} \cdot P_{2}} & 0 \\ 0 & {G_{4} \cdot P_{4}} \end{bmatrix}_{224 \times 224}}} & \left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack \\ {{C = {{\begin{bmatrix} O_{1} \\ O_{3} \end{bmatrix}_{224 \times 1} + {j \cdot \begin{bmatrix} O_{2} \\ O_{4} \end{bmatrix}_{224 \times 1}}} = {C_{I} + {j \cdot C_{Q}}}}}{{R_{I} + {j \cdot R_{Q}}} = {\begin{bmatrix} V_{O\; 1} \\ V_{O\; 3} \end{bmatrix} + {j \cdot {\begin{bmatrix} V_{O\; 2} \\ V_{O\; 4} \end{bmatrix}.}}}}} & \left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack \end{matrix}$

Ideal ADC outputs S_(I) and S_(Q) can be retrieved as the following Equation 11.

S _(I) +jS _(Q) =M _(I) ⁻¹{circle around (×)}_(c) R _(I) +j·M _(Q) ⁻¹{circumflex over (×)}_(c) R _(Q) +C′  [Equation 11]

wherein C′ may be represented by the following Equation 12.

C′=−M _(I) ⁻¹

C _(I) −j·M _(Q) ⁻¹

C _(Q)  [Equation 12]

Let the 2× oversampled transmitted QPSK signal vectors through X and Y polarizations are X_(I,Q) and Y_(I,Q), respectively. X_(I,Q) and Y_(I,Q) can be written in 112×1 matrix as given by the following Equation 13.

X _(I,Q) =[X _(I,Q) ¹ [n]X _(I,Q) ¹ [n]X _(I,Q) ² [n]X _(I,Q) ² [n] . . . X _(I,Q) ⁵⁶ [n]] ^(T)

Y _(I,Q) =[Y _(I,Q) ¹ [n]Y _(I,Q) ¹ [n]Y _(I,Q) ² [n]Y _(I,Q) ² [n] . . . X _(I,Q) ⁵⁶ [n]] ^(T)

wherein

X _(I,Q) ^(i) [n]=X _(I,Q)[56n+i] and Y _(I,Q) ^(i) [n]=y _(I,Q)[56n+i].  [Equation 13]

Note that x_(I,Q) and y_(I,Q) denote transmitted symbols. CD in the radix-112 matrix format may be represented by the following Equation 14.

CD _(ij) [n]=cd[112n+i−j]. i,j=1 . . . 112  [Equation 14]

wherein cd[n] denotes the 2× oversampled complex impulse response of CD.

The first-order polarization mode dispersion (PMD) can also be written as a radix-112 matrix, as the following Equation 15.

$\begin{matrix} {{{P\; M\; D} = \begin{bmatrix} {\cos \; \alpha \times I^{\theta_{x}}} & {{- \sin}\; \alpha \times I^{\theta_{y}}} \\ {\sin \; \alpha \times I^{\theta_{x}}} & {\cos \; \alpha \times I^{\theta_{y}}} \end{bmatrix}}{{I_{ij}^{\theta_{x,y}} = {\delta \left\lbrack {{112\; n} + i - j - \theta_{x,y}} \right\rbrack}},i,{j = {1\mspace{14mu} \ldots \mspace{14mu} 112}}}} & \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack \end{matrix}$

PMD adds a phase delay of _ to each polarization and attenuates the transmitted signal by rotating the polarization angle by α. The combined channel dispersion matrix H including CD and PMD may be represented by the following Equation 16.

$\begin{matrix} {H_{224 \times 224} = {\begin{bmatrix} \lbrack{CD}\rbrack_{112 \times 112} & 0 \\ 0 & \lbrack{CD}\rbrack_{112 \times 112} \end{bmatrix} \otimes_{c}\left\lbrack {P\; M\; D} \right\rbrack_{224 \times 224}}} & \left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack \end{matrix}$

Let matrix denote the combined transmitted signal as given by the tollowing Equation 17.

$\begin{matrix} {\hat{D} = \begin{bmatrix} {X_{I} + {j \cdot X_{Q}}} \\ {Y_{I} + {j \cdot Y_{Q}}} \end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack \end{matrix}$

Because the received dispersed signal at the input of the ADC may be related to D as the following Equation 18.

[{circumflex over (D)}] _(224×1) =[H] _(224|×224) ⁻¹

S _(I) +j·[H] _(224×224) ⁻¹

S _(Q)  [Equation 18]

the transmitted data matrix {circumflex over (D)} can be retrieved from nonideal outputs of the ADCs by using Equation 11 and Equation 18 as the following Equation 19.

{circumflex over (D)}=Â

R _(I) +j·{circumflex over (B)}

R _(Q) +Ĉ″  [Equation 19]

wherein Â and {circumflex over (B)} may be represented by the following Equation 20 and Ĉ″ may be represented by the following Equation 21.

Â _(224×224) =H ⁻¹

M _(I) ⁻¹ , {circumflex over (B)} _(224×224) =H ⁻¹

M _(Q) ⁻¹  [Equation 20]

C″=H ⁻¹

C′  [Equation 21]

However, because the received signal is 2× eversampled, only half of the components in {circumflex over (D)} must be equalized. Therefore, D may be represented by the following Equation 22.

D=A

R _(I) +j·B

R _(Q) +C″  [Equation 22]

Wherein A and B may be represented by the following Equation 23. Also, D and C″ may be represented by the following Equation 24.

$\begin{matrix} {{A_{112 \times 224} = \begin{bmatrix} {\hat{A}}_{1} \\ {\hat{A}}_{3} \\ \ldots \\ {\hat{A}}_{223} \end{bmatrix}},{B_{112 \times 224} = \begin{bmatrix} {\hat{B}}_{1} \\ {\hat{B}}_{3} \\ \ldots \\ {\hat{B}}_{223} \end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack \\ {{D_{112 \times 1} = \begin{bmatrix} {{X_{I}^{1}\lbrack n\rbrack} + {j\; {X_{Q}^{1}\lbrack n\rbrack}}} \\ \ldots \\ {{X_{I}^{56}\lbrack n\rbrack} + {j\; {X_{Q}^{56}\lbrack n\rbrack}}} \\ {{Y_{I}^{1}\lbrack n\rbrack} + {j\; {Y_{Q}^{1}\lbrack n\rbrack}}} \\ \ldots \\ {{Y_{I}^{56}\lbrack n\rbrack} + {j\; {Y_{Q}^{56}\lbrack n\rbrack}}} \end{bmatrix}},{C_{112 \times 1}^{''} = \begin{bmatrix} {\hat{C}}_{1}^{''} \\ {\hat{C}}_{3}^{''} \\ \ldots \\ {\hat{C}}_{223}^{''} \end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack \end{matrix}$

Equation 22 can be realized with a MIMO equalizer and the mismatches in the parallel ADCs and the channel dispersion can be compensated for simultaneously by adapting the coefficients of A, B, and C″ using a least mean square (LMS) algorithm.

The MIMO equalizer doesn't require a front-end equalizer for the compensation of the gain and phase mismatches of an optical hybrid. It is because the MIMO equalizer can compensate for such non-idealities together with the gain and phase mismatches of an ADC.

FIG. 4 shows a block diagram of the MIMO equalizer according to an exemplary embodiment of the present invention. A₁˜A₁₁₂ denote sub-equalizers and A_(1,1)˜A_(1,224) denote the coefficients of A_(I), as given by the following Equation 25.

$\begin{matrix} {{A = \begin{bmatrix} A_{1} \\ A_{2} \\ \ldots \\ A_{112} \end{bmatrix}}{A_{1} = \begin{bmatrix} A_{1,1} & {A_{1,2}\mspace{14mu} \ldots} & A_{1,224} \end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 25} \right\rbrack \end{matrix}$

FIG. 5 shows a section of hardware utilized for the equalization of two transmitted symbols through X and Y polarizations according to an exemplary embodiment of the present invention. The length of a fractionally spaced sub-equalizer (FSE) is L. The coefficients of the FSE may satisfy the following Equation 26.

XA ₁[112×n+k]=A _(1,k) [n]k=1 . . . 112,

YA ₁[112×n+k−112]=A _(1,k) [n]k=113 . . . 224,

XB ₁[112×n+k]=B _(1,k) [n]k=1 . . . 112,

XB ₁[112×n+k−|112]=B _(1,k) [n]k=113 . . . 224.

The total number of parallelized sub-equalizers for one polarization is 56, as 112 parallelized ADCs are oversampling the received signal by a factor of 2. The total number of real value multiplications required for the reconstruction of two transmitted symbols X_(IQ) and X_(IQ) in the MIMO equalizer is L×8×2=16×L as shown in the following Table 1 (a). Note that the factor of 2 is multiplied because the filter coefficients are complex numbers.

FIG. 6 shows the block diagrams of one sub-equalizer in a conventional parallel FIR equalizer for 112 Gb/s DP-QPSK coherent systems. L₁ and L₂ denote the length of the HR filter taps required for the compensation of CD and PMD, respectively. L₁ and L₂ should satisfy L₁+L₂=L given that the dispersion compensation capabilities of conventional and MIMO equalizers are equal. CD and PMD are compensated for separately and each FSE receives complex samples from two ADCs. Because the PMD equalization process is performed sequentially, the CD equalizer should maintain a oversampling ratio to minimize the SNR penalty in the PMD compensator. Note that both filter coefficients and sampled ADC outputs are complex in this case. The total number of real-value multiplications required for the reconstruction of two transmitted symbols X_(IQ) and Y_(IQ) in the conventional equalizer is (L₁×2×2+L₂×4)×4=16×(L₁+L₂) which is identical to 16×L as shown in the following Table 1 (b). Consequently, the MIMO equalizer has no size penalty in terms of multiplier counts over conventional equalizers and does not require area-hungry ADC calibration circuits.

TABLE 1 real-value FSE length parallel multiplier factor Total (a) L 8 2 1 16 × L (b) L₁ 9 4 2 16 (L₁ + L₂) L₂ 4 4 1

B. ADC Nonlinearity Compensation

SAR ADC is considered as most suitable type for coherent optical communication. It has been shown that the nonlinearity of a SAR ADC can be compensated for using a digital-domain signal processing method. Unlike the previous work which uses a reference ADC for calibration, the proposed DA MIMO equalizer can compensate for the nonlinearity of a SAR ADC and channel dispersion simultaneously by using the estimated output mean square error. Because this compensation process is an interpretation rather than a calibration scheme to deal with the non-linearity, a certain amount of SNR penalty at the final output can exist.

A SAR ADC may consist of a capacitance array for digital-to-analog conversion (DAC), a comparator for the decision, and a digital logic block for the DAC control. FIG. 7 shows a block diagram of a five-bit switched-capacitor SAR-ADC according to an exemplary embodiment of the present invention. When switch SW is connected during the sampling period, the input signal V_(in) is sampled at the bottom plate of the capacitor array and V₁˜V₆ become Vin. The switch SW is turned off after the sampling period and V₁˜V₆ are connected to V₁˜V₆. The voltage change at V_(x), defined as ΔV_(x), is related to the voltage changes at V₁˜V₆ as given by the following Equation 27:

$\begin{matrix} {\frac{\Delta \; V_{x}}{\Delta \; V_{i}} = {\frac{C_{i}}{C_{tot} + C_{p}} = {{K_{i}\mspace{31mu} i} \in \left\{ {1\mspace{14mu} \ldots \mspace{14mu} 6} \right\}}}} & \left\{ {{Equation}\mspace{14mu} 27} \right\rbrack \end{matrix}$

where ΔVi denotes the voltage change at V_(i), C_(tot)=τ_(i=1) ⁶C_(i) and C_(p) is the parasitic capacitance. The total voltage change AY, may be represented by the following Equation 28.

ΔV _(x)=Σ_(i=1) ⁶ K _(i) ΔV _(i)=Σ_(i=1) ⁶ K _(i)(−V _(R) −V _(in))  [Equation 23]

The capacitor array receives digital codes d₁˜d₅ from the digital logic block and adds the corresponding analog voltage to the sampled input signal V_(in) using a charge redistribution process. The digital logic block uses a binary search algorithm to find the digital code which takes V_(x) to the sub-LSB level. V_(x) is related to the digital code d₁˜d₅, V_(in) and V_(R) as given by the following Equation 29.

$\begin{matrix} \begin{matrix} {V_{x} = {{\sum\limits_{i = 1}^{6}{K_{i}\left( {{- V_{R}} - V_{i\; n}} \right)}} + {\sum\limits_{i = 1}^{5}{{d_{i} \cdot 2}\; K_{i}V_{R}}}}} \\ {= {{\sum\limits_{i = 1}^{5}{\left( {{2\; d_{i}} - 1} \right)K_{i}V_{R}}} - {K_{0}V_{R}} - {K_{tot}V_{i\; n}}}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 29} \right\rbrack \end{matrix}$

wherein K_(tot) may be represented by the following Equation 30.

$\begin{matrix} {K_{tot} = {{\sum\limits_{i = 1}^{6}K_{i}} = \frac{C_{tot}}{C_{tot} + C_{p}}}} & \left\lbrack {{Equation}\mspace{14mu} 30} \right\rbrack \end{matrix}$

Hence, the input signal V_(in) may be represented by the following Equation 31.

$\begin{matrix} \begin{matrix} {V_{i\; n} = {K_{tot}^{- 1}\left( {{\sum\limits_{i = 1}^{5}{\left( {{2\; d_{i}} - 1} \right)K_{i}V_{R}}} - {K_{0}V_{R}} - V_{x}} \right)}} \\ {= {K_{tot}^{- 1}\left( {{\sum\limits_{i = 1}^{5}{\left( {{2\; d_{i}} - 1} \right)K_{i}V_{R}}} - {K_{0}V_{R}} - \left( {V_{Q} + V_{os}} \right)} \right)}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 31} \right\rbrack \end{matrix}$

wherein K_(os) V_(Q) are the comparator offset and the quantization noise, respectively. Equation 31 shows that the input signal V_(in) can be accurately reconstructed from digital codes d₁˜d₅ by multiplying adequate coefficients with each bit and adding a proper offset in a SAR ADC.

The corrected output signal V_(O,corr) of a five-bit SAR ADC can be written as the following Equation 32.

V _(O,corr)=Σ_(i=1) ⁵ a _(i) ×d _(i) +o _(c)  [Equation 32]

wherein a_(i) and o_(c) are the coefficient and offset for the correction, respectively. The correction factors a_(i) and o_(c) were retrieved using a slow-but-accurate reference ADC in the earlier work. However, a_(i) can be combined with the filter coefficients of the digital FIR filter by using a DA scheme as shown in FIG. 8, and adjusted adaptively by an LMS algorithm. FIG. 3 shows conceptual block diagram based on merging coefficients for nonlinearity compensation with those for MIMO equalization according to an exemplary embodiment of the present invention.

FIG. 9 shows a block diagram of a sub-equalizer of the DA MIMO equalizer according to an exemplary embodiment of the present invention. The coefficient in the DA MIMO equalizer is related to that of the MIMO equalizer A_(j,k) ^(i) in the Equation 25 as given by the following Equation 33.

$\begin{matrix} {{A_{j,k}^{i} = {a_{i}^{k} \times A_{j,k}}},{i = {1\mspace{14mu} \ldots \mspace{14mu} 5}},{j = {1\mspace{14mu} \ldots \mspace{14mu} 112}},{k = {1\mspace{14mu} \ldots \mspace{14mu} 224}}} & \left\lbrack {{Equation}\mspace{14mu} 33} \right\rbrack \end{matrix}$

where the correction factor a_(i) ^(k) may satisfy the following Equation 34.

V _(O1,corr) ^(k)=Σ_(i=1) ⁵ a _(i) ^(k) ×d _(i) ^(k) +o _(c) ^(k),

V _(O3,corr) ^(k)=Σ_(i=)1⁵ a _(i) ^(k+)112×d _(i) ^((k+112))+o _(c) ^(k)+112  [Equation 34]

In summary, channel dispersion and ADC nonlinearity can be compensated for simultaneously by multiplying different FIR filter coefficients A_(j,k) ¹ . . . A_(j,k) ⁵ with each ADC output bit and combining them at the output. The offset correction factor o_(c) is implemented in the last stage of the DA MIMO equalizer by subtracting the average values of output E_(i,j) from the recovered symbols. This process aligns the center of the QPSK signal space of each sub-equalizer to the origin.

Iv. A Variable-Precision Distributed Arithmetic (VPDA) MIMO Equalizer

The VPDA MIMO equalizer reduces the dynamic power consumption of the DA MIMO equalizer by using only the minimum required resolution for the equalization of each dispersed symbol. FIG. 10 shows the equalized QPSK symbols under different ADC resolutions in a two-dimensional signal space according to an exemplary embodiment of the present invention. The magnitude of the standard deviations σ_(k) and the resulting BER are inversely proportional to the resolution of the ADC. However, equalized symbols located distant from decision thresholds in the signal space can be correctly determined with high probability even under low ADC resolutions. In other words, the required resolution for the equalization of each dispersed symbol is different. FIG. 11 shows a conceptual block diagram of the proposed variable-precision architecture embedded in the DA framework according to an exemplary embodiment of the present invention. The proposed pipelined equalizer begins the computation from the most significant bit (MSB) towards the least significant bit (LSB) sequentially. Range checkers are inserted between the DA-based subequalizers described in FIG. 9 to determine whether further computation with a higher resolution is required. In case a partially equalized symbol is located outside the suspicious region in the signal space (see FIG. 11), a final decision is made and no further computations are performed. In contrast, if the partial result is within the suspicious region, the precision of the partial result is increased by one bit and the location of the equalized symbol in the signal space is rechecked. Thus, the average resolution for the equalization can be reduced significantly and hence the dynamic power consumption can also be decreased.

The estimated power ratio of the VPDA MIO equalizer over the DA MIMO equalizer may be approximately represented by the following Equation 35.

$\begin{matrix} {\frac{P_{VPDA}}{P_{DA}} \approx {\sum\limits_{k = 1}^{5}{\frac{1}{5} \times {\Pr \left( {En}_{k} \right)}}}} & \left\lbrack {{Equation}\mspace{14mu} 35} \right\rbrack \end{matrix}$

where P_(VPDA) and P_(DA) are the power consumptions of the VPDA MIMO and the DA MIMO equalizers, respectively, and Pr(En_(k)) denotes the probability that a single bit equalizer at each resolution step is being enabled as shown in FIG. 11.

The probability Pr(En_(k)) is determined by both SNR and the area of suspicious region at each resolution step. Proper selection of the suspicious region in the signal space is crucial for the VPDA MIMO equalizer because premature inaccurate decisions caused by insufficient area of the suspicious regions increase BER penalty. Therefore, the design target for the dynamic power minimization of the VPDA MIMO equalizer is to minimize the average ADC resolution by minimizing the area of the suspicious regions without a significant BER penalty. The transition rate, defined as the probability of an equalized symbol being in the suspicious region in the k^(th) stage depends on both the size of the suspicious region in the current stage and the transition rate in the previous stages. Thus, the size of the suspicious regions should be determined sequentially from MSB to LSB. The suspicious region of the first stage, Sus₁, is determined based on the BER while assuming that the suspicious regions in the subsequent stage are infinite and that no extra bit-error occurs from the variable-precision architecture. The estimated BER at the output of the first stage may be represented by the following Equation 36.

BER=Pr(bit error ∩P ₁ ^(c))+Pr(bit error ∩P ₁ ∩P ₅ ^(c))  [Equation 36]

wherein P_(k) denotes the set of events that the output symbol in the k^(th) stage, Sym_(k)=(X_(k), Y_(k)), is in the suspicious region and P_(k) ^(c) denotes the complementary set of P_(k). In case symbol (1, 1) is transmitted, Pr (bit error ∩P_(k) ^(c)) can be expanded as the following Equation 37.

$\begin{matrix} {{\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{k}^{c}} \right)} = {{{\Pr \left( {{bit}\mspace{14mu} {error}} \middle| E_{k}^{x} \right)} \cdot {\Pr \left( E_{k}^{x} \right)}} + {{\Pr \left( {{bit}\mspace{14mu} {error}} \middle| E_{k}^{y} \right)} \cdot {\Pr \left( E_{k}^{y} \right)}} + {{\Pr \left( {{bit}\mspace{14mu} {error}} \middle| E_{k}^{xy} \right)} \cdot {\Pr \left( E_{k}^{xy} \right)}}}} & \left\lbrack {{Equation}\mspace{14mu} 37} \right\rbrack \end{matrix}$

wherein E_(k) ^(x), E_(k) ^(y) and E_(k) ^(xy) are the sets of erroneous events, as given by the following Equation 38.

E _(k) ^(x)={Sym_(k) εERR _(x) ^(k)}

E _(k) ^(y)={Sym_(k) εERR _(y) ^(k)}

E _(k) ^(xy)={Sym_(k) εERR _(xy) ^(k)}  [Equation 38]

wherein ERR_(x) ^(k), ERR_(y) ^(k) and ERR_(xy) ^(k) denote the areas shown in FIG. 12. FIG. 12 shows erroneous regions in the QFSK signal space when (1, 1) is sent according to an exemplary embodiment of the present invention. Note that the conditional probabilities in the Equation 37 are be represented by the following Equation 39.

$\begin{matrix} {{{\Pr \left( {{bit}\mspace{14mu} {error}} \middle| E_{k}^{x} \right)} = \frac{1}{2}}{{\Pr \left( {{bit}\mspace{14mu} {error}} \middle| E_{k}^{y} \right)} = \frac{1}{2}}{{\Pr \left( {{bit}\mspace{14mu} {error}} \middle| E_{k}^{xy} \right)} = 1}} & \left\lbrack {{Equation}\mspace{14mu} 39} \right\rbrack \end{matrix}$

Because the two-dimensional probability density function of Sym_(k) is symmetric with respect to Y_(k)=X_(k), the Equation 37 can be simplified to the following Equation 4.

$\begin{matrix} \begin{matrix} {{\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{k}^{c}} \right)} = {{\frac{1}{2} \cdot {\Pr \left( E_{k}^{x} \right)}} + {\frac{1}{2} \cdot {\Pr \left( E_{k}^{y} \right)}} + {\Pr \left( E_{k}^{xy} \right)}}} \\ {= {{\Pr \left( E_{k}^{x} \right)} + {\Pr \left( E_{k}^{xy} \right)}}} \\ {= {\Pr \left( {E_{k}^{x}\bigcup E_{k}^{xy}} \right)}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 40} \right\rbrack \end{matrix}$

Because the noise in each equalized symbol is a linear combination of independent and identically distributed (i.i.d) additive noises caused by quantization and limited input signal-to-noise ratio in each sample, it can be assumed to show a Gaussian distribution according to the central limit theorem. Then, the probabilities Pr(E_(k) ^(x)) and Pr(E_(k) ^(xy)) are given by the following Equation 41 and Equation 42.

$\begin{matrix} \begin{matrix} {{\Pr \left( E_{k}^{x} \right)} = {\int_{- \infty}^{- {Sus}_{1}}{\int_{{Sus}_{1}}^{\infty}{{{g\left( {x,1,\sigma_{k}} \right)} \cdot {g\left( {y,1,\sigma_{k}} \right)}}{x}{y}}}}} \\ {= {{Q\left( \frac{{Sus}_{1} + 1}{\sigma_{k}} \right)} \cdot {Q\left( \frac{{Sus}_{1} - 1}{\sigma_{k}} \right)}}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 41} \right\rbrack \\ \begin{matrix} {{\Pr \left( E_{k}^{xy} \right)} = {\int_{- \infty}^{- {Sus}_{1}}{\int_{- \infty}^{- {Sus}_{1}}{{g\left( {x,1,\sigma_{k}} \right)} \cdot}}}} \\ {{g\left( {y,1,\sigma_{k}} \right){x}{y}}} \\ {= {Q\left( \frac{{Sus}_{1} + 1}{\sigma_{k}} \right)}^{2}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 42} \right\rbrack \end{matrix}$

wherein σ_(k) is the standard deviation of the noise at k^(th) stage and g(x,μ,σ) is a Gaussian function, as given by the following Equation 43.

$\begin{matrix} {{g\left( {x,\mu,\sigma} \right)} = {\frac{1}{\sqrt{2\pi}\sigma} \cdot ^{\frac{{({x - \mu})}^{2}}{2\sigma^{2\;}}}}} & \left\lbrack {{Equation}\mspace{14mu} 43} \right\rbrack \end{matrix}$

Because P_(k) is symmetrical with respect to Y_(k)=X_(k), Pr(bit error ∩P_(m)∩P_(n) ^(c)), m≦n can be written using Equation 40 as the following Equation 44.

$\begin{matrix} \begin{matrix} {{\Pr \left( {\left\{ {{{bit}\mspace{14mu} {error}}\bigcap P_{n}^{c}} \right\}\bigcap P_{m}} \right)} = {\Pr \left( {\left\{ {E_{n}^{x}\bigcup E_{n}^{xy}} \right\}\bigcap P_{m}} \right)}} \\ {= {{\Pr \left( {E_{n}^{x}\bigcap P_{m}} \right)} + {\Pr \left( {E_{n}^{xy}\bigcap P_{m}} \right)}}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 44} \right\rbrack \end{matrix}$

Let N_(m,n) be the noise added to the equalized output signal when ADC resolution is reduced from n bit to in bit. N_(m,n) can be modeled by a Gaussian distribution of N(0,√{square root over (σ_(m) ²−σ_(n) ²)}). Then, Pr(E_(n) ^(x)∩P_(m)) may be given by the following Equation 45.

$\begin{matrix} \begin{matrix} {{\Pr \left( {E_{n}^{x}\bigcap P_{m}} \right)} = {\Pr \begin{pmatrix} {\left\{ {X_{n} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {{Sus}_{n} < Y_{n}} \right\}} \\ {\bigcap\begin{Bmatrix} {\left\{ {{- {Sus}_{m}} < X_{m} < {Sus}_{m}} \right\}\bigcup} \\ \left\{ {{- {Sus}_{m}} < Y_{m} < {Sus}_{m}} \right\} \end{Bmatrix}} \end{pmatrix}}} \\ {= {\Pr \begin{pmatrix} {\left\{ {X_{n} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {{Sus}_{n} < Y_{n}} \right\}} \\ {\bigcap\begin{Bmatrix} \begin{Bmatrix} {{- {Sus}_{m}} < {N_{m,n}^{x} +}} \\ {X_{n} < {Sus}_{m}} \end{Bmatrix} \\ {\bigcup\begin{Bmatrix} {{- {Sus}_{m}} < {N_{m,n}^{y} +}} \\ {Y_{n} < {Sus}_{m}} \end{Bmatrix}} \end{Bmatrix}} \end{pmatrix}}} \\ {= {\Pr \begin{pmatrix} {\left\{ {X_{n} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {{Sus}_{n} < Y_{n}} \right\}} \\ {\bigcap\begin{Bmatrix} \begin{Bmatrix} {{{- {Sus}_{m}} - X_{n}} <} \\ {N_{m,n}^{x} < {{Sus}_{m} - X_{n}}} \end{Bmatrix} \\ {\bigcup\begin{Bmatrix} {{{- {Sus}_{m}} - Y_{n}} <} \\ {N_{m,n}^{y} < {{Sus}_{m} - Y_{n}}} \end{Bmatrix}} \end{Bmatrix}} \end{pmatrix}}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 45} \right\rbrack \end{matrix}$

The Equation 45 is changed to the following Equation 46.

$\begin{matrix} {{\Pr \left( {E_{n}^{x}\bigcap P_{m}} \right)} = {{\Pr \begin{pmatrix} {\left\{ {X_{n} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {{Sus}_{n} < Y_{n}} \right\}\bigcap} \\ \left. \left\{ {{{- {Sus}_{m}} - X_{n}} < N_{m,n}^{x} < {{Sus}_{m} - X_{n}}} \right\} \right\} \end{pmatrix}} + {\Pr \begin{pmatrix} {\left\{ {X_{n} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {{Sus}_{n} < Y_{n}} \right\}\bigcap} \\ \left. \left\{ {{{- {Sus}_{m}} - Y_{n}} < N_{m,n}^{y} < {{Sus}_{m} - Y_{n}}} \right\} \right\} \end{pmatrix}} - {\Pr \begin{pmatrix} {\left\{ \left| {X_{n} < {- {Sus}_{n}}} \right. \right\}\bigcap\left\{ {{Sus}_{n} < Y_{n}} \right\}\bigcap} \\ \begin{Bmatrix} \left\{ {{{- {Sus}_{m}} - X_{n}} < N_{m,n}^{x} < {{Sus}_{m} - X_{n}}} \right\} \\ {\bigcap\left\{ {{{- {Sus}_{m}} - Y_{n}} < N_{m,n}^{y} < {{Sus}_{m} - Y_{n}}} \right\}} \end{Bmatrix} \end{pmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 46} \right\rbrack \end{matrix}$

Similarly, Pr(E_(n) ^(x)∩P_(m)) may be represented by the following Equation 47.

$\begin{matrix} {{\Pr \left( {E_{n}^{xy}\bigcap P_{m}} \right)} = {{\Pr \begin{pmatrix} {\left\{ {X_{n} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {Y_{n} < {- {Sus}_{n}}} \right\}\bigcap} \\ \left. \left\{ {{{- {Sus}_{m}} - X_{n}} < N_{m,n}^{x} < {{Sus}_{m} - X_{n}}} \right\} \right\} \end{pmatrix}} + {\Pr \begin{pmatrix} {\left\{ {X_{m} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {Y_{n} < {- {Sus}_{n}}} \right\}\bigcap} \\ \left. \left\{ {{{- {Sus}_{m}} - Y_{n}} < N_{m,n}^{y} < {{Sus}_{m} - Y_{n}}} \right\} \right\} \end{pmatrix}} - {\Pr \begin{pmatrix} {\left\{ {X_{n} < {- {Sus}_{n}}} \right\}\bigcap\left\{ {Y_{n} < {- {Sus}_{n}}} \right\}\bigcap} \\ \begin{Bmatrix} \left\{ {{{- {Sus}_{m}} - X_{n}} < N_{m,n}^{x} < {{Sus}_{m} - X_{n}}} \right\} \\ {\bigcap\left\{ {{{- {Sus}_{m}} - Y_{n}} < N_{m,n}^{y} < {{Sus}_{m} - Y_{n}}} \right\}} \end{Bmatrix} \end{pmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 47} \right\rbrack \end{matrix}$

Assuming that the random variables X_(n), Y_(n), N_(m,n) ^(x) and N_(m,n) ^(y) are independent, Equation 45 becomes the following Equation 48.

$\begin{matrix} {{\Pr \left( {E_{n}^{x}\bigcap P_{m}} \right)} = {{\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( x_{n} \right)}{{x_{n}} \cdot {Q\left( \frac{{- 1} + {Sus}_{n}}{\sigma_{n}} \right)}}}} + {\int_{{Sus}_{u\; 1}}^{\infty}{{F_{m,n}\left( y_{n} \right)}{{y_{n}} \cdot {Q\left( \frac{1 + {Sus}_{n}}{\sigma_{n}} \right)}}}} - {\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( x_{n} \right)}{{x_{n}} \cdot {\int_{{Sus}_{n}}^{\infty}{{F_{m,n}\left( y_{n} \right)}{y_{n}}}}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 48} \right\rbrack \end{matrix}$

wherein F_(m,n)(x) may be defined in the following Equation 49.

$\begin{matrix} {{F_{m,n}(x)} = {\left( {{Q\left( \frac{{- {Sus}_{m}} - x}{\sqrt{\sigma_{m}^{2} - \sigma_{n}^{2}}} \right)} - {Q\left( \frac{{Sus}_{m} - x}{\sqrt{\sigma_{m}^{2} - \sigma_{n}^{2}}} \right)}} \right) \cdot {g\left( {x,1,\sigma_{n}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 49} \right\rbrack \end{matrix}$

Similarly, Pr(E_(n) ^(xy)∩P_(m)) may be given by the following b b th Equation 50.

$\begin{matrix} {{\Pr \left( {E_{n}^{xy}\bigcap P_{m}} \right)} = {{{\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( x_{n} \right)}{{x_{n}} \cdot {Q\left( \frac{1 + {Sus}_{n}}{\sigma_{n}} \right)}}}} + {\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( y_{n} \right)}{{y_{n}} \cdot {Q\left( \frac{1 + {Sus}_{n}}{\sigma_{n}} \right)}}}} - {\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( x_{n} \right)}{{x_{n}} \cdot {\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( y_{n} \right)}{y_{n}}}}}}}} = {{2 \cdot {\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( x_{n} \right)}{{x_{n}} \cdot {Q\left( \frac{1 + {Sus}_{n}}{\sigma_{n}} \right)}}}}} + \left( {\int_{- \infty}^{- {Sus}_{n}}{{F_{m,n}\left( x_{n} \right)}{x_{n}}}} \right)^{2}}}} & \left\lbrack {{Equation}\mspace{14mu} 50} \right\rbrack \end{matrix}$

The addition of the Equation 48 and the Equation 50 renders the conditional probability of Pr({bit error ∩P_(n) ^(c)}∩P_(m)). Then, the relationship between Sus₁ and BER can be achieved from (36) and the minimum Sus₁ satisfying a BER target can be chosen. The relationship between BER and Sus₂ with the minimum Sus₁ value chosen above may be the following Equation 51.

$\begin{matrix} {{BER} = {{\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{1}^{c}} \right)} + {\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{1}\bigcap P_{2}^{c}} \right)} + {\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{1}\bigcap P_{2}\bigcap P_{5}^{c}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 51} \right\rbrack \end{matrix}$

Pr(bit error∩P₁∩P₂∩P₅ ^(c)) can be simplified to Pr(bit error∩P₂∩P₅ ^(c)) given that Sus₂ is smaller than Sus₁; thus, P₁∩P₂≈P₂. In general, the relationship between BER and Sus_(k) with a predetermined minimum Sus₁ . . . Sus_(k−1) may be given by the following Equation 52.

$\begin{matrix} {{{{BER} = {{\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{1}^{c}} \right)} + {\sum\limits_{i = 2}^{k}{\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{i - 1}\bigcap P_{i}^{c}} \right)}} + {\Pr \left( {{{bit}\mspace{14mu} {error}}\bigcap P_{k}\bigcap P_{5}^{c}} \right)}}}\mspace{20mu} {{k = 2},3,4}}\mspace{25mu}} & \left\lbrack {{Equation}\mspace{14mu} 52} \right\rbrack \end{matrix}$

provided that the following Equation 53.

Pr(P ₁ ∩P ₂ . . . P _(k))≈Pr(P _(k))  [Equation 53]

wherein Pr(P_(k)) is derived in the following Equation 54. Finally, the dynamic power reduction ratio of the VPDA MIMO equalizer can be estimated by using the Equation 35 because Pr(P_(k))=Pr(En_(k+1)).

$\begin{matrix} \begin{matrix} {{\Pr \left( P_{k} \right)} = {\Pr \begin{pmatrix} {\left\{ {{- {Sus}_{k}} < X_{k} < {Sus}_{k}} \right\}\bigcup} \\ \left\{ {{- {Sus}_{k}} < Y_{k} < {Sus}_{k}} \right\} \end{pmatrix}}} \\ {= {{\Pr \left( \left\{ {{- {Sus}_{k}} < X_{k} < {Sus}_{k}} \right\} \right)} +}} \\ {{{\Pr \left( \left\{ {{- {Sus}_{k}} < Y_{k} < {Sus}_{k}} \right\} \right)} -}} \\ {{\Pr\left( {\left\{ {{- {Sus}_{k}} < X_{k} < {Sus}_{k}} \right\}\bigcap\left\{ {{- {Sus}_{k}} < Y_{k} < {Sus}_{k}} \right.} \right.}} \\ {= {{2 \cdot \left( {{Q\left( \frac{{- 1} - {Sus}_{k}}{\sigma_{k}} \right)} - {Q\left( \frac{{- 1} + {Sus}_{k}}{\sigma_{k}} \right)}} \right)} -}} \\ {\left( {{Q\left( \frac{{- 1} - {Sus}_{k}}{\sigma_{k}} \right)} - {Q\left( \frac{{- 1} + {Sus}_{k}}{\sigma_{k}} \right)}} \right)^{2}} \end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 54} \right\rbrack \\ {\mspace{20mu} {{k = {1\mspace{14mu} \ldots \mspace{14mu} 4}}\mspace{20mu} {{\Pr \left( P_{5} \right)} = 0}}} & \; \end{matrix}$

V. Simulation

The simulation setup of the VPDA MIMO equalizer is shown in FIG. 13. The channel and optical components are modeled using OPTSIM and the equalizer is simulated using MATLAB. Chromatic dispersion up to 1360 ps/nm, equivalent to 801 cm in SMF-28 single mode fiber at a wavelength of 1550 nm, without PMD is considered as the channel impairment. Hence, 56 Gbps OPSK data transmitted through a single polarization is used for the verification of the proposed VPDA MIMO architecture for simplicity. The bandwidth of the optical filter at the receiver is set to 60 GHz and third-order Bessel filters with a bandwidth of 19 GHz are used. 201322 randomly generated bits are used for the BER simulation, where the BER target is 10′. The data sampled by the two nonlinear 56 GS/s ADCs is fed into a conventional equalizer, a DA MIMO equalizer and a VPDA MIMO equalizer for comparison. The coefficients of each equalizer are adapted using an LMS algorithm. A total of 16×7×2=224 different nonlinear characteristics of SAR ADCs are randomly generated for two 56 GS/s ADCs by randomizing the capacitance mismatches and comparator offsets. The standard deviations of the capacitances are set proportional to √{square root over (area)} as given by the following Equation 55.

σ_(C) _(k) =α×√{square root over (2^(5-k))}×C ₆

C _(k)=2^(5-k) ×C ₆ , k=1 . . . 5

wherein α is a process parameter. In this simulation α is set to 0.4. The standard deviation of the comparator offset is set to 40 mV and the full scale of the ADC is set to ±400 mV_(peak-peak). Total 112 randomly generated nonlinear characteristics of a single ADC are shown in FIG. 14. The gain and sampling phase mismatches between 16 track and hold circuits are randomly generated with standard deviations of 0.2V/V and 4 ps(0.22 UI), respectively. The precisions of the coefficients of the DA and VPDA MIMO FIR equalizers are set to 8 bit in order to suppress the penalty due to the finite precision below 0:1 dB at a BER of 10⁻³. The simulated values of σ_(k) in FIG. 10 for the different ADC resolutions are summarized in the following Table. 2. The relationships between BER and the range of the suspicious regions for each VPDA MIME stage from the Equation 52 are plotted as solid lines in FIG. 15. FIG. 15 shows BER increment with respect to the size of the suspicious regions for each stage according to an exemplary embodiment of the present invention. The hollow circle markers in the figure denote the simulated BER points.

TABLE 2 resolution 1-bit 2-bit 3-bit 4-bit 5-bit σ_(k) 0.7262 0.5045 03931 0.3534 03407

The analytic results (lines) discussed in section IV closely match the simulation results (hollow circle). The normalized suspicious regions with respect to the QPSK signal space for each level of ADC precision from MSB to LSB are set to 1.82, 0.86, 0.44 and 0.19 as shown in FIG. 15 considering the BER penalty. The following Table 3 summarizes the analytical and simulated enable rate of the equalizer at each resolution step with the suspicious regions selected in the above.

TABLE 3 Pr(En_(k)) Pr(En₂) Pr(En₃₎ Pr(En₄) Pr(En₅) Eq.(54) 0.983 0.625 0.147 0.021 simulation 0.985 0.624 0.143 0.020

The estimated dynamic power ratio of the proposed VPDA MIMO equalizer over the DA MIMO equalizer is 0.55 from the Equation 35 and thus 45% of dynamic power consumption can be reduced. FIG. 16 shows the simulated OSNR vs. BER graphs for three different equalizers according to an exemplary embodiment of the present invention. The VPDA MIMO equalizer with the dynamic power reduction of 45% shows a negligible OSNR penalty compared to the DA MIMO equalizer. The VPDA MIMO equalizer subsequent to the non-ideal ADCs shows a 0.5 dB worst-case OSNR penalty compared to the ideal five bit ADCs followed by an ideal equalizer at a BER of 10⁻³. In contrast, the conventional equalizer with an identical nonideal ADCs shows an OSNR penalty of more than 2.5 dB at a BER of 10⁻³. The adaptation speed of the VPDA MIMO equalizer is 56 times slower than that of a conventional parallel equalizer because one adaptation engine sequentially sets the coefficients of 56 parallel sub-equalizers. FIG. 17 shows the simulated tracking performance, of the VPDA MIMO equalizer under the rotation of the polarization plane at a rate of 50 rad/s according to an exemplary embodiment of the present invention. Such a rotational rate is considered severe in 80 km applications. Variable precision scheme is disabled for simplicity. SNR is set to 13.5 dB and the initial coefficients of the VPDA MIMO equalizer are set to 0. The clock frequency is 500 MHz and the BER is measured every 3.6 us. The estimated time constant of the adaptation engine is less than 2 ms and the VPDA MIMO equalizer demonstrates negligible BER penalty.

VI. Summary

A power-and-area efficient BER-aware VDPA MIMO architecture for a 112 Gb/s DP-QPSK coherent receiver is presented. The VPDA MEMO equalizer achieves 45% dynamic power reduction compared to conventional FIR equalizers and does not require area-hungry analog domain calibration circuits for the ADC.

The exemplary embodiments according to the present invention may be recorded in computer-readable media including program instructions to implement various operations embodied by a computer. The media may also include, alone or in combination with the program instructions, data files, data structures, and the like. The media and program instructions may be those specially designed and constructed for the purposes of the present invention, or they may be of the kind well-known and available to those having skill in the computer software arts.

It will be apparent to those skilled in the art that various modifications and variation can be made in the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention cover the modifications and variations of this invention provided they come within the scope of the appended claims and their equivalents. 

What is claimed is:
 1. A variable-precision distributed arithmetic (VPDA) multi-input multi-output (MIMO) equalizer connected to outputs of a plurality of analog-to-digital converters (ADCs) based on time-interleaved successive approximation registers, the VPDA MIMO equalizer comprising: a plurality of sub-equalizers classified into a first sub-equalizers group and a second sub-equalizers group, wherein each of sub-equalizers included in the first sub-equalizers group is connected to outputs of a first ADC group and each of sub-equalizers included in the second sub-equalizers group is connected to outputs of a second ADC group; and a decision unit configured to determine output signals using outputs of the plurality of sub-equalizers, wherein one sub-equalizer included in the first sub-equalizers group relates to other one sub-equalizer included in the second sub-equalizers group, and one output signal among the output signals is determined by the decision unit based on outputs of the one sub-equalizer and the other one sub-equalizer.
 2. The VPDA MIMO equalizer of claim 1, wherein each of the plurality of sub-equalizers comprises a plurality of additional-equalizers for distributed arithmetic, and each additional-equalizers corresponds to one of bits according to a resolution of the ADCs.
 3. The VPDA MIMO equalizer of claim 2, wherein one additional-equalizer of the one sub-equalizer relates to other one additional-equalizer of the other one sub-equalizer, and the one output signal is determined by the decision unit based on an output of the one additional-equalizer and an output of the other one additional-equalizer.
 4. The VPDA MIMO equalizer of claim 2, wherein outputs of the ADCs corresponding to i-th bit of the bits is inputted to i-th additional-equalizers of the plurality of sub-equalizers.
 5. The VPDA MIMO equalizer of claim 2, wherein each of the plurality of additional-equalizers comprises a plurality of finite impulse responses (FIR) filters.
 6. The VPDA MIMO equalizer of claim 5, wherein one part of the plurality of FIR filters is connected to outputs of one ADC of the first ADC group (or the second ADC group), and another part of the plurality of FIR filters is connected to outputs of other one ADC of the first ADC group (or the second ADC group).
 7. The VPDA MIMO equalizer of claim 2, wherein the plurality of additional-equalizers configured to computing outputs according to a sequence from the most significant bit (MSB) of the bits towards the least significant bit (LSB) of the bits.
 8. The VPDA MIMO equalizer of claim 8, further comprising: a range checker configured to determine whether further computing outputs for next bit of the sequence is required based on a equalized symbol determined by outputs for present bit and a decision threshold.
 9. The VPDA MIMO equalizer of claim 1, wherein coefficients with each of bits according to a resolution of the ADCs are combined with filter coefficients of FIR filters by at least one of a distributed arithmetic (DA) scheme and a least mean square (LMS) algorithm.
 10. The VPDA MIMO equalizer of claim 9, wherein each of outputs of the ADCs is multiplied with the combined coefficients and each of outputs of the FIR filters is combined.
 11. A dual-polarization quadrature phase-shift-keying (DP-QPSK) receiver comprising: a plurality of analog-to-digital converters (ADCs) based on time-interleaved successive approximation registers; a variable-precision distributed arithmetic (VPDA) multi-input multi-output (MIMO) equalizer connected to outputs of the ADCs, wherein the VPDA MIMO equalizer comprises a plurality of sub-equalizers, and each of sub-equalizers comprises a plurality of additional-equalizers; and a decision unit configured to determine output signals using outputs of the additional-equalizers.
 12. The DP-QPSK receiver of claim 11, wherein a number of additional-equalizers included in one sub-equalizer corresponds to a number of bits determined by a resolution of the ADCs, and outputs of the ADCs corresponding to i-th bit of the bits is inputted to i-th additional-equalizers of the plurality of sub-equalizers.
 13. The DP-QPSK receiver of claim 11, wherein each of the plurality of additional-equalizers comprises a plurality of finite impulse responses (FIR) filters.
 14. The DP-QPSK receiver of claim 13, wherein one part of the plurality of FIR filters is connected to outputs of one ADC and another part of the plurality of FIR filters is connected to outputs of other one ADC.
 15. The DP-QPSK receiver of claim 11, wherein the plurality of additional-equalizers configured to compute outputs according to a sequence from the most significant bit (MSB) towards the least significant bit (LSB), and the MSB and the LSB are determined by a resolution of the ADCs. 